I was just wondering whether current research has dealt with storing DNA of people or animals (e.g. rodents) and observing how it changed, especially for cases where it eventually resulted in cancer. Something along the lines of this, but I'd like to know what research has actually been done so far.

2 Answers
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I tried to comment but what I wrote is too long, so here it is as an answer of sorts.

If I understand the question, you are asking: has anyone done a prospective study where they store the DNA of individuals and then later, when some of these individuals get cancer, have a look for mutations that are associated with that cancer. In fact this is done all of the time, but it isn't necessary to store any DNA because the cancer patients will have the normal sequence in all of their non cancerous tissues - only the cancer cells will have the relevant mutations. See here for a recent example of this type of study that made the news.

Your BBC link is to a story in which the genomes of people will be analysed to determine if there are genetic factors which predispose them to cancer (and other diseases), but these won't be the mutations that actually create the cancerous cells.